Difference between a solenoid and a stack of loops

The B field at the center of a stack of N circular loops each carrying current I is
B = mu0 * N * I / (2a)
where a is the radius of the loop, derived using Biot-Savart law.

The B field inside a solenoid is
B = mu0 * N * I / l
where l is the length of the solenoid, derived using Ampere's law.

Yet everywhere I searched it is always stated that a solenoid can be thought of as a stack of circular loops. Then why are the results different? Why can't I use Ampere's law on the stack of loops to get mu0 * N * l?

It is also stated that in a solenoid the B field at the ends of the solenoid is 1/2 of the B field inside the solenoid, or
B = mu0 * N * I / (2l)

Could that be it? That is, the B field calculated using Biot-Savart law for stack of loops is the same as that for a solenoid B field but ONLY at the ends of the solenoid?

I am citing Figures 28.14 and 28.24 in University Physics by Young and Freedman, 13th edition.

No. There is nothing about the equation for the B field of a stack of rings being valid only for a short length. The number of loops is N and can be as large as I want. Moreover, the solenoid does not have to be infinite for the equation to be valid.

Of course I recognize the quantity "a" in the loop equation is the radius of the loop and "l" in the solenoid equation is the length of the solenoid. I am trying to reconcile these 2 formulas.

Staff: Mentor

No. There is nothing about the equation for the B field of a stack of rings being valid only for a short length. The number of loops is N and can be as large as I want. Moreover, the solenoid does not have to be infinite for the equation to be valid.

Just check where these equations come from, and which assumptions were made to derive them. If the stack of rings is allowed to have a variable length, this length would have to appear in the formula.

In the same way, I recognize the formula for solenoids, and it uses the approximation that the solenoid is very long relative to its diameter.